The pin group of a definite form maps onto the orthogonal group, and each component is simply connected: it double covers the orthogonal group. The pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are.[note 1]

In terms of the standard forms, O(n, 0) = O(0,n), but Pin(n, 0) and Pin(0, n) are not isomorphic. Using the "+" sign convention for Clifford algebras (where ), one writes

There are as many as eight different double covers of O(p, q), for p, q ≠ 0, which correspond to the extensions of the center (which is either C2 × C2 or C4) by C2. Only two of them are pin groups—those that admit the Clifford algebra as a representation. They are called Pin(p, q) and Pin(q, p) respectively.

Every connectedtopological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group. For a disconnected topological group, there is a unique universal cover of the identity component of the group, and one can take the same cover as topological spaces on the other components (which are principal homogeneous spaces for the identity component) but the group structure on other components is not uniquely determined in general.

The Pin and Spin groups are particular topological groups associated to the orthogonal and special orthogonal groups, coming from Clifford algebras: there are other similar groups, corresponding to other double covers or to other group structures on the other components, but they are not referred to as Pin or Spin groups, nor studied much.

Recently, Andrzej Trautman[note 3] found the set of all 32 inequivalent double covers of O(p) x O(q), the maximal compact subgroup of O(p, q) and an explicit construction of 8 double covers of the same group O(p, q).

The group structure on Spin(V) (the connected component of determinant 1) is already determined; the group structure on the other component is determined up to the center, and thus has a ±1 ambiguity.

The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker (Spin(V) → SO(V)), and the two pin groups are named accordingly. Explicitly, a reflection has order 2 in O(V), r2 = 1, so the square of the preimage of a reflection (which has determinant one) must be in the kernel of Spin±(V) → SO(V), so , and either choice determines a pin group (since all reflections are conjugate by an element of SO(V), which is connected, all reflections must square to the same value).

Concretely, in Pin+, has order 2, and the preimage of a subgroup {1, r} is C2 × C2: if one repeats the same reflection twice, one gets the identity.

In Pin−, has order 4, and the preimage of a subgroup {1, r} is C4: if one repeats the same reflection twice, one gets "a rotation by 2π"—the non-trivial element of Spin(V) → SO(V) can be interpreted as "rotation by 2π" (every axis yields the same element).

In 2 dimensions, the distinction between Pin+ and Pin− mirrors the distinction between the dihedral group of a 2n-gon and the dicyclic group of the cyclic group C2n.

In Pin+, the preimage of the dihedral group of an n-gon, considered as a subgroup Dihn < O(2), is the dihedral group of an 2n-gon, Dih2n < Pin+(2), while in Pin−, the preimage of the dihedral group is the dicyclic group.

The resulting commutative square of subgroups for Spin(2), Pin+(2), SO(2), O(2) – namely C2n, Dih2n, Cn, Dihn – is also obtained using the projective orthogonal group (going down from O by a 2-fold quotient, instead of up by a 2-fold cover) in the square SO(2), O(2), PSO(2), PO(2), though in this case it is also realized geometrically, as "the projectivization of a 2n-gon in the circle is an n-gon in the projective line".

In 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups: